Theory of Machines Course # 1

Transcription

1 Theory of Machines Course # 1 Ayman Nada Assistant Professor Jazan University, KSA. arobust@tedata.net.eg March 29, 2010

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5 Chapter 1 INTRODUCTION 1.1 Introduction Mechanisms may be de ned as the division of machine design concerned with the kinematic design of linkages, cams, gears, and gear trains. Kinematic design is the design on the basis of motion requirements in contrast to the design on basis of strength requirements. The study of mechanisms and machines is an applied science used to understand the relation between the motions of their elements and the forces producing these motions within some geometrical constraints. With the continuous advances in designing instruments and automated systems, the study of mechanisms becomes of great importance. This chapter is concerned with the study of simple mechanisms topological structure, kinematic diagram, inversions, mobility index, degrees of freedom, geometric constraints, and geometry of motion. The functions of many important mechanisms are also included. These items are important for the study of mechanism motion. The chapter is organized in ve main sections: (a) Basic de nitions and mechanism elements (b) Kinematic chain, kinematic diagram and mechanism inversions (c) Examples of important mechanisms (d) Mobility index, degrees of freedom, geometric constraints, redundancy, and exibility (e) Mechanism topology and geometry of motion The chapter also includes 4 solved examples and ends by a set of problems. 1.2 Basic De nitions To better understand the mechanics of mechanisms, it is necessary to keep in mind the following de nitions: Mechanism: A combination of rigid and/or exible bodies connected in such away to do work and there are de nite constrained relative motions between them.

6 4 CHAPTER 1. INTRODUCTION Structure: The same de nition of mechanism, but its purpose is not to do work and there is no relative motion between its parts. Machine: An arrangement of parts and/or mechanisms for doing work and there are constrained relative motions between its parts. Statics: The part of mechanics, which deals with the action of forces on bodies at rest. Kinematics: Study of motion without reference to the forces causing the motion. Kinetics: Relates the action of forces on bodies to their resulting motions. Dynamics: The part of mechanics, which deals with the action of forces on bodies in motion. Mechanics: Deals with the action of forces on bodies at rest and in motion. Figure 1.1: Mechanism Figure 1.2: A 3D truss structure

7 1.3. MECHANISM ELEMENTS AND CLASSIFICATION Mechanism Elements and Classi cation A Mechanism is composed of three main elements: links, pairing elements, and a drive or drives. The links are connected together with kinematic pairs, called joints, to permit their constraints relative motions. A mechanism is normally driven through a transmission system, which may include belts, ropes, chains, and/or gears, by a motor. Mechanism links may be rigid, uidics, or exible. For the sake of simplicity, links are assumed rigid and joints have perfect geometry with no clearance through out this text. In many mechanisms springs are used for restoring forces and do not a ect their kinematics. Mechanism Links: Links through out the text are considered rigid and the number of joints on each link gives its type. In other words, a binary link is that having 2 joints (Fig. 1.3.a), a ternary link is that having 3 joints (Fig. 1.3.b) and a quaternary link is that having 4 joints (Fig. 1.3.c). A well-known ternary link is the bell crank shown in Fig Other names are given to mechanism links such as: input link, output link, driving link, driven link, initial link, frame, base, bar, rocker, coupler, sliding block, slider, guide, crosshead, ram, connecting rod, and many other names. This class of links makes the so-called linkage mechanisms such as the crank-slider mechanism (Fig. 1.5) and the 4-bar linkage (Fig. 1.6). Figure 1.3: Types of links (a) binary, (b) ternary, (c) quaternary Cams and followers are another class of mechanism links, which make the so-called camfollower mechanisms as those shown in Fig Mechanism Joints: There are two types of connecting pairing elements: lower pairs and higher pairs. Lower pairs have surface contact between mating elements and higher pairs have line or point contact. The contact surface of a shaft in a bearing and that of the wrist pin joining the piston and connecting rod as well as the surface between the piston and the cylinder are some examples of lower pairs. Lower pairs include spherical (S), revolute (R), cylindrical (C) and prismatic (P) joints which represented in Fig.1.8. The contact between a cam and a follower or between two meshing gear teeth is examples of higher pairs. Table 1.2: Classi cation of linkage joints Mechanism Classi cations: There are three types of mechanisms: planar, spherical, and spatial. In planar mechanisms, all particles describe plane parallel curves in space while in spatial mechanisms there is no restrictions on the relative motions of particles. In spherical mechanisms, each link has some point, which remains stationary as the linkage moves. The stationary points of all links lie at the same location in space. Hook s or universal joint

8 6 CHAPTER 1. INTRODUCTION Figure 1.4: Bell crank Figure 1.5: Slider crank mechanism Figure 1.6: Four Bar Mechanism

9 1.3. MECHANISM ELEMENTS AND CLASSIFICATION 7 Figure 1.7: Cam-follower mechanism Figure 1.8: Mechanical Joints

10 8 CHAPTER 1. INTRODUCTION Table 1.1: Generalized coordinates and position of an arbitrary point used in automobile is an example of spherical mechanisms. If spherical mechanisms have only revolute joints they are called spherical linkages. Mechanisms may form closed loops, open loops or combination. Crank-slider mechanism and 4-bar linkage are of closed loop type while robot arms are of open loop. Another mechanism classi cation based on the type of their links is linkage, cam-follower, and gearset or gear train mechanisms. In practice a mechanism may be a combination of all these types such as the engine mechanism. 1.4 Examples of Important Mechanisms Slider-Crank Mechanisms Slider-Crank Mechanism: The sketch of linkage arranged as shown in Fig.1.9 is known as the slider-crank mechanism. Link 1 is a stationary base or a frame, link 2 is the crank, link 3 is the connecting rod, and link 4 is the slider. The line of slider stroke passes through the center of crank rotation. O set Slider-Crank Mechanism: The slider crank can be o set as shown in Fig This o set produces a quick return motion for the slider. However, the amount of quick return is very slight, the mechanism would be only used where space is limited. Scotch Yoke Mechanism: This mechanism is sketched in Fig It consists of the same elements as slider-crank mechanism and is early used in steam pumps and in computing machines as a harmonic generator. Recently, it is used as a mechanism on a test machine to produce vibrations Four-Bar Linkage The 4-bar linkage consists of 4 pin-connected rigid links as shown in Fig There are many types and names of the 4-bar linkage depending on the mechanism dimensions. These include double crank, crank-rocker, drag link, double-rocker, and crossover-piston or changepoint mechanisms. For crank-rocker type, link 1 is the frame, which is stationary, link 2 is the crank, which makes complete revolutions, link 3 is the coupler, and link 4 is the rocker,

11 1.4. EXAMPLES OF IMPORTANT MECHANISMS 9 Figure 1.9: Slider-Crank Mechanisms Figure 1.10: Slider-Crank Mechanisms with o set

12 10 CHAPTER 1. INTRODUCTION Figure 1.11: Scotch Yoke Mechanism which performs the desired task. This simple mechanism is important, as it is the base of many other mechanisms. For these reasons it will be studied in all details through out the text. Figure 1.12: Four Bar Mechanism Quick-Return Mechanisms Several types of quick-return mechanisms QRM are in use in machine tools. The QRM give quick return strokes and slow cutting strokes for constant angular velocities of the driving cranks. These mechanisms are combinations of simple linkages such as the 4-bar linkage and the slider-crank mechanism. An inversion of the slider crank in combination with the conventional slider crank is also used. All known QRM are described after.

13 1.4. EXAMPLES OF IMPORTANT MECHANISMS 11 Figure 1.13: Crank-Shaper Mechanism

14 12 CHAPTER 1. INTRODUCTION 1.5 MOBILITY OF MECHANISMS The mobility of a mechanism is its number of degrees of freedom. This translates into a number of independent input motions leading to a single follower motion. A single unconstrained link (Figure 1.14.a) has three DOF in planar motion: two translational and one rotational. Thus, two disconnected links (Figure 1.14.b) will have six DOF. If the two links are welded together (Figure 1.14.c), they form a single link having three DOF. A revolute joint in place of welding (Figure 1.14.d) allows a motion of one link relative to another, which means that this joint introduces an additional (to the case of welded links) DOF. Thus, the two links connected by a revolute joint have four DOF. One can say that by connecting the two previously disconnected links by a revolute joint, two DOF are eliminated. Similar considerations are valid for a prismatic joint. Figure 1.14: Various con gurations of links with two revolute joints Since the revolute and prismatic joints make up all low-pair joints in planar mechanisms, the above results can be expressed as a rule: a low-pair joint reduces the mobility of a mechanism by two DOF. These results are generalized in the following formula, which is called Kutzbach s criterion of mobility M = 3(n 1) 2j 1 j 2 where n is the number of links, j 1 is the number of low-pair joints, and j 2 is the number of high-pair joints. Note that 1 is subtracted from n in the above equation to take into account that the mobility of the frame is zero. In Figure 1.15 the mobility of various con gurations of connected links is calculated. All joints are low-pair ones. Note that the mobility of the links in Fig.1.15.a is zero, which means that this system of links is not a mechanism, but a structure. At the same time, the system of interconnected links in Fig.1.15.d has mobility 2, which means that any two links can be used as input links (drivers) in this mechanism. Look at the e ect of an additional link on the mobility. This is shown in Fig.1.16, where a four-bar mechanism (Figure 1.16.a) is transformed into a structure having zero mobility (Figure 1.16.b) by adding one link, and

15 1.6. GRASHOF S LAW FOR A FOUR-BAR MECHANISM 13 then into a structure having negative mobility (Figure 1.16.c) by adding one more link. The latter is called an overconstrained structure. Figure 1.15: Mobility of various con gurations of connected links:(a) n = 3; j 1 = 3; j 2 = 0; m = 0; (b) n = 4; j 1 = 4; j 2 = 0; m = 1;(c) n = 4; j 1 = 4; j 2 = 0; m = 1;(d) n = 5; j 1 = 5; j 2 = 0; m = 2: In compound mechanisms, there are links with more than two joints. Kutzbach s criterion is applicable to such mechanisms provided that a proper account of links and joints is made. Consider a simple compound mechanism shown in Fig.1.17, which is a sequence of two fourbar mechanisms. In this mechanism, joint B represents two connections between three links. In other words, it should be taken into account that there are, in fact, two revolute joints at B. The axes of these two joints may not necessarily coincide. According to Kutzbach s formula M = = GRASHOF S law for a Four-Bar mechanism The fourbar linkage has been shown above to be the simplest possible pin-jointed mechanism for single degree of freedom controlled motion. It also appears in various disguises such as the slider-crank and the cam-follower. It is in fact the most common and ubiquitous device used in machinery. It is also extremely versatile in terms of the types of motion which it can generate Simplicity is one mark of good design. The fewest parts that can do the job will usually give the least expensive and most reliable solution. Thus the fourbar linkage should be among the rst solutions to motion control problems to be investigated. The Grashof condition is a very simple relationship which predicts the rotation behavior or rotatability of a fourbar linkage s inversions based only on the link lengths. Let S =length of shortest link

16 14 CHAPTER 1. INTRODUCTION Figure 1.16: E ect of additional links on mobility: (a)m = 1; (b)m = 0; (c)m = 1: Figure 1.17: An example of a compound mechanism with coaxial joints at B

17 1.6. GRASHOF S LAW FOR A FOUR-BAR MECHANISM 15 L = length of longest link P = length of one remaining link Q =length of other remaining link Then if : S + L P + Q the linkage is Grashof and at least one link will be capable of making a full revolution with respect to the ground plane. This is called a Class I kinematic chain. If the inequality is not true, then the linkage is non-grashof and no link will be capable of a complete revolution relative to any other link. This is a Class II kinematic chain. Note that the above statements apply regardless of the order of assembly of the links. That is, the determination of the Grashof condition can be made on a set of unassembled links. Whether they are later assembled into a kinematic chain in S, L, P, Q, or S, P, L, Q or any other order, will not change the Grashof condition. The motions possible from a fourbar linkage will depend on both the Grashof condition and the inversion chosen. The inversions will be de ned with respect to the shortest link. The motions are: For the Class I case, S + L < P + Q Ground either link adjacent to the shortest and you get a crank-rocker, in which the shortest link will fully rotate and the other link pivoted to ground will oscillate. Ground the shortest link and you will get a double-crank, in which both links pivoted to ground make complete revolutions as does the coupler. Ground the link opposite the shortest and you will get a Grashof double-rocker, in which both links pivoted to ground oscillate and only the coupler makes a full revolution. Determine the mobility index and the degrees of freedom of each of the plane mechanisms shown in Fig All joints are of R type.

18 Chapter 2 Kinematic Analysis of Mechanisms There are various methods of performing kinematic analysis of mechanisms, including graphical, analytical, and numerical. The choice of a method depends on the problem at hand and on available computational means. Kinematic analysis of a mechanical system means the computation, at any time instant, of the mechanism con gurations, positions, displacements, linear velocities and accelerations of its interesting points as well as the angular velocities and accelerations of its links. This chapter deals with the kinematics of planar linkage mechanisms using graphical and analytical methods. More emphasis is given to analytical methods in order to simplify the use of computers for mechanism animation and simulation. The analysis of the 4-bar linkage, slider-crank mechanism, and the shaper quick return mechanism is used through out the chapter to illustrate the used methods for linkage kinematic analysis. The chapter ends with a set of interesting problems.

19 2.1. COORDINATE SYSTEMS AND VECTOR REPRESENTATION Coordinate Systems and Vector Representation For planar mechanisms, two coordinate systems are used: rectangular (x; y) and polar (r; ) as shown in Fig. ( ). The choice of the coordinate system is arbitrary and must be selected to suit the situation. After de ning the mechanism working space, reference frame, and time instant, its kinematic analysis is possible using graphical or analytical methods. In this text, only planar mechanisms are considered and vectors are represented either in Cartesian coordinates as x and y components or in polar coordinates by its magnitude r and phase angle or by complex numbers. Rotation in planar motion is always represented by a vector normal to the plane of motion i.e. the z-axis. Figure 2.1: Coordinate System Position Vector: The vector ~r or ~r P de ning the absolute position of point P; Fig. ( ) is represented in polar coordinates by its magnitude and phase angle or by complex numbers as: ~r = ~r p = r\ = r e j = r (cos + j sin ) where j is the imaginary number. Velocity Vector: The rst time derivative of the position vector r de nes the absolute velocity of point P; ~v P as:

20 18 CHAPTER 2. KINEMATIC ANALYSIS OF MECHANISMS Figure 2.2: Velocity vector ~v P = ~r p = _r e j + j r _ e j = _r e j + r _ e j (+=2) If r is constant in magnitude, the absolute velocity of point P is given by: ~v P = r _ e j (+=2) = r!\ ( + =2) where, _ is the angular velocity of the vector OP, _ =!: Acceleration Vector: The second time derivative of the position vector ~r de nes the ~a P = ~r p = ~a =r e j (+=2) + r!e j (+) = r e j (+=2) + r! 2 e j (+) = r \ ( + =2) + r! 2 \ ( + ) = ~a P ~a = ~a t + ~a r

21 2.1. COORDINATE SYSTEMS AND VECTOR REPRESENTATION 19 where ~a t = r \ ( + =2) ~a r = r! 2 \ ( + ) Figure 2.3: Acceleration vectors Where is the angular acceleration of the vector OP, and ~a t and ~a r are respectively the acceleration components.

22 20 CHAPTER 2. KINEMATIC ANALYSIS OF MECHANISMS Example 1 Slider Crank Mechanism The general linkage con guration and terminology for a slider-crank linkage with o est are shown in Figure (2.4). The link lengths and the values of 2,! 2 and 2 are de ned in the table. For the row(s) assigned, draw the linkage to scale and nd the velocities of the pin joints A and B and the velocity of slip at the sliding joint using a graphical method. Figure 2.4: Con guration and terminology row Link 2 Link 3 O set 2! 2 2 f e g

23 2.1. COORDINATE SYSTEMS AND VECTOR REPRESENTATION 21 (f.) Results: Figure 2.5: Example : (f.) Position row v B! 3 a B 3 v f. 2:5 50 = 125 B=A r 3 = 0:650 a = 2:307 2: = 2470 B=A t 13 r 3 = 6: = 466:15 13

24 22 CHAPTER 2. KINEMATIC ANALYSIS OF MECHANISMS Figure 2.6: Example : (f.) Velocity & Acceleration

25 2.1. COORDINATE SYSTEMS AND VECTOR REPRESENTATION 23 (e.) Results: Figure 2.7: Example : (e.) Position row v B! 3 a B 3 v e. 1: = 191 B=A r 3 = 1:77100 a = 8:85 3: = 6620 B=A t 20 r 3 = 4: =

26 24 CHAPTER 2. KINEMATIC ANALYSIS OF MECHANISMS Figure 2.8: Example : (e.) Velocity & Acceleration